A new method to detect projective equivalences and symmetries of rational 3D curves

Detecting Affine Equivalences Between Implicit Planar Algebraic Curves

“…), which by Theorem 3 is a necessary condition for affine equivalence. The following result, analogous to Theorem 7 in [4] and which can be proved, using Theorem 3, in a similar way, shows that this condition is also sufficient.…”

“…The description of step (i) is analogous to Section 3.2 in [4]. Thus, here we focus on the main ideas, and refer the interested reader to [4] for details and proofs. Going back to Eq.…”

“…In the last years, the problem has been also addressed in the Computer Algebra field, and this paper follows this trend. Examples of papers where this question has been studied are [2,4,7]; for other related papers, the interested reader can check the bibliographies of [2,4,7]. These papers address the problem for rational curves, i.e.…”

“…While [2,7] provide solutions for this problem, with different strategies, for curves in any dimension and using as a fundamental tool polynomial system solving, the paper [4] addresses the question only for space rational curves, but employing bivariate factoring as an alternative to solving polynomial systems; this leads to better timings and performance. To do this, in [4] two rational invariants, i.e. two functions rationally depending on the parametrizations to be studied which stay invariant under projective transformations, are found and used.In this paper we generalize the ideas of [4] in three different ways.…”

“…In order to solve this problem, we will make use of the following result, which corresponds to a similar result used in [4,7], adapted to our case. We recall here that a Möbius transformation is a transformation…”